In this setting, deformations are restricted a priori according to the distance between the obstacle Ψ and the body Ω. Let g: ΓC → R denote the corresponding function which measures the distance at each pointx∈ΓC along the unit outer normal vectorn. Then, enforcing a non-penetration condition along the normalncorresponds to requiring that
φn≤g a.e. on ΓC. (2.20)
The Signorini contact condition was first analyzed in [97] and has been extensively studied ever since, see, e.g., [20, 53, 56]. Apparently, Condition (2.20) is a strong sim-plification since it is far from obvious that contact will always occur along the outer normaln. The validity of such conditions depends on the geometry and the initial set-ting. Thus, the question arises whether the Signorini condition sufficiently describes the underlying contact problem. At least for small displacements, and if the boundary segment ΓC and the obstacle Ψ are very close and essentially parallel, cf. [56, Chapter 2], Condition (2.20) is a suitable approximation. A more detailed discussion of this topic can be found in [25, 73]. For settings involving large deformations, and in particular for multi-body problems, closest point projections are widely applied to model contact conditions, see, e.g., [59, 60, 109, 110].
Additional complications arise in the case of multi-body contact problems. These prob-lems require specialized discretization schemes that apply to non-matching grids such as the mortar method, see, e.g., [12, 50, 81, 91, 108, 110, 111]. A further option to model contact conditions is the normal compliance method, which is studied in Chapter 3.
In summary, the study of more sophisticated contact constraints and geometries is still the subject of current research and remains beyond the scope of this work. A more de-tailed discussion about modeling contact constraints, though not for nonlinear elasticity, can be found in [73, 91]. Applying contact constraints in nonlinear elasticity has been discussed in [81, 110, 111]. For an overview of contact problems in general, the reader is referred to [43, 56].
2.4.2 Contact problems in hyperelasticity
Recalling Theorem 2.16, we know that under certain smoothness assumptions minimizers of the total energy functionalI describe deformations of a body. In contrast to before, the total energy functional
I :Y ×U →R∪ {∞}
is a function of the deformationyand of the applied forceu. The corresponding boundary forceudacting on the deformed segment Γd,N is a dead load by assumption. As a result, u does not depend on the deformationy. Consequently, we write
I(y, u) :=
Z
Ω
Wˆ(x,∇y(x))dx− Z
ΓN
y(x)u(x)ds. (2.21)
This notation is later required to analyze optimal control problems combined with elas-ticity. We also define the splitting
I(y, u) =Istrain(y)−Iout(y, u) with
Istrain(y) :=
Z
Ω
Wˆ(x,∇y(x))dx and Iout(y, u) :=
Z
ΓN
y(x)u(x)ds. (2.22) Further restrictions on the function spaces and the stored energy function ˆW are required since we have to ensure that the resulting energy minimization problem is well-posed and that our setting corresponds to the physical model elaborated in the previous sections.
The necessary properties are summarized in the following assumption.
Assumption 2.31. LetWˆ : Ω×M3+ →Rbe the stored energy function of a hyperelastic material and let (p, r, s, q, q0) ∈]1,∞[5 be fixed with p≥2, s≥ p−1p , and r >1. Assume that the following properties hold.
1. Polyconvexity: For almost all x∈Ω, there exists a convex function W(x,·,·,·) :M3×M3×]0,∞[→R such that
Wˆ(x, M) =W(x, M,CofM,detM) for allM ∈M3+. Further, the function
W(·, M,Cof M,detM) : Ω→R is measurable for all (M,CofM,detM)∈M3×M3×]0,∞[.
2. For almost all x∈Ω, the implication
detM →0+⇒Wˆ(x, M)→ ∞ holds.
3. The sets of admissible deformations are defined by
Ac:={y∈W1,p(Ω), Cof∇y∈Ls(Ω), det∇y∈Lr(Ω),
y= id a.e. on ΓD, det∇y >0 a.e. inΩ, y3 ≥0 a.e. on ΓC} and
A:={y ∈W1,p(Ω), Cof∇y∈Ls(Ω), det∇y ∈Lr(Ω), y = id a.e. on ΓD, det∇y >0 a.e. in Ω}. 4. Coerciveness: There exist constants a∈R andb >0 such that
Wˆ(x, M)≥a+b(kMkp+kCof Mks+|detM|r) for allM ∈M3+.
2.4. CONTACT PROBLEMS 31 5. The exponents q, q0 ∈]1,∞[satisfy q−1+q0−1 = 1,
q0< 2p
3−p for p <3, and q0 <∞ for p≥3.
6. For the zero boundary force uz ∈ U, the identity mapping id : Ω → Ω satisfies id∈argminv∈AcI(v, uz) with I(id, uz)<∞.
The first assumption states the polyconvexity of the stored energy function ˆW. This property is necessary to compensate for the non-convexity of ˆW and show the existence of solutions to hyperelastic problems.
Assumptions 2.31(2) and 2.31(4) describe the physical behavior of the material for large strains. In this context, Assumption 2.31(2) corresponds to the condition that com-pressing a given volume to zero requires an infinite amount of work. Assumption 2.31(4) is a sharper version of the coerciveness property stated in Assumption 2.24 since the exponents p,s, andr satisfy stronger restrictions here.
Further, from Assumption 2.31(3), we obtain the admissible set of deformationsAc for the obstacle constrained case. Later, we introduce a regularization approach for the contact constraints which allows us to operate on the relaxed admissible set A.
The definition of the admissible setA and the first assumption ensure that the integral Istrain(y) is well-defined for all y∈ A, cf. [15, Proof of Theorem 7.7-1].
The coerciveness of the strain energyIstrain is implied by Assumption 2.31(4), see, e.g., [15, Proof of Theorem 7.3-2].
Assumption 2.31(5) is a technical requirement to apply the H¨older inequality and ensure the existence of compact trace operators.
Finally, Assumption 2.31(6) guarantees that A and Ac are not empty. Also, for some fixed boundary forceu∈U, we obtain infv∈AcI(v, u)<∞and infv∈AI(v, u)<∞. Note here that the non-emptiness of the admissible sets essentially depends on the boundary conditions imposed on ΓD. Since admissible deformations coincide with the identity mapping on ΓD, the non-emptiness of the admissible sets A and Ac follows directly.
However, this implication no longer holds for general boundary conditions imposed on ΓD. In those cases, the non-emptiness of the admissible sets is assumed a priori. This is basically an assumption on the functionyD, which describes the boundary conditions on ΓD.
For a detailed discussion of these requirements, the reader is referred to [15, Chapter 7] and [18]. We assume that these assumptions are satisfied throughout this work.
With Assumption 2.31 at hand, modeling contact problems corresponds to a well-posed minimization problem.
Definition 2.32 (Contact problem). Let u∈U be a fixed boundary force applied to a hyperelastic body. Further, the deformation of the body is restricted by Condition (2.19).
Then, the resulting deformationy satisfies y∈argmin
v∈Ac
I(v, u). (2.23)
With Theorem 2.16 in mind, the question arises whether the minimization problem (2.23) also corresponds to solving some kind of equilibrium equations. However, in the case of contact problems, the relation can only be derived formally.
Theorem 2.33. Let y be a sufficiently smooth solution to the energy minimization problem
y∈argmin
v∈Ac
I(v, u).
Then, y formally solves the following boundary value problem:
−divTe(∇y(x)) = 0 for allx∈Ω, Te(∇y(x))n(x) =u(x) for allx∈ΓN,
y(x) = id(x) for allx∈ΓD, y3(x)≥0 for allx∈ΓC,
Te(∇y(x))n(x) = 0 if x∈ΓC and y3(x)>0,
Te(∇y(x))n(x) =λ(x)nd(y(x)) withλ(x)≤0 if x∈ΓC and y3(x) = 0.
Proof. For the proof, see [15, Proof of Theorem 5.3-1].